Problem: Factor the following expression: $-5$ $x^2$ $-12$ $x$ $-4$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(-4)} &=& 20 \\ {a} + {b} &=& & & {-12} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $20$ and add them together. The factors that add up to ${-12}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-2}$ and ${b}$ is ${-10}$ $ \begin{eqnarray} {ab} &=& ({-2})({-10}) &=& 20 \\ {a} + {b} &=& {-2} + {-10} &=& -12 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-5}x^2 {-2}x {-10}x {-4} $ Group the terms so that there is a common factor in each group: $ ({-5}x^2 {-2}x) + ({-10}x {-4}) $ Factor out the common factors: $ x(-5x - 2) + 2(-5x - 2) $ Notice how $(-5x - 2)$ has become a common factor. Factor this out to find the answer. $(-5x - 2)(x + 2)$